It is known that, for every constant $kgeq 3$, the presence of a $k$-clique (a complete subgraph on $k$ vertices) in an $n$-vertex graph cannot be detected by a monotone boolean circuit using fewer than $Omega((n/log n)^k)$ gates. We show that, for every constant $k$, the presence of an $(n-k)$-clique in an $n$-vertex graph can be detected by a monotone circuit using only $O(n^2log n)$ gates. Moreover, if we allow unbounded fanin, then $O(log n)$ gates are enough.
@InProceedings{andreev_et_al:DagSemProc.06111.22, author = {Andreev, Alexander E. and Jukna, Stasys}, title = {{Very Large Cliques are Easy to Detect}}, booktitle = {Complexity of Boolean Functions}, pages = {1--7}, series = {Dagstuhl Seminar Proceedings (DagSemProc)}, ISSN = {1862-4405}, year = {2006}, volume = {6111}, editor = {Matthias Krause and Pavel Pudl\'{a}k and R\"{u}diger Reischuk and Dieter van Melkebeek}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/DagSemProc.06111.22}, URN = {urn:nbn:de:0030-drops-6092}, doi = {10.4230/DagSemProc.06111.22}, annote = {Keywords: Clique function, monotone circuits, perfect hashing} }
Feedback for Dagstuhl Publishing